Mathematics in Computer Science

, Volume 1, Issue 3, pp 487–505 | Cite as

Dichotomy Results for Fixed-Point Existence Problems for Boolean Dynamical Systems

Article

Abstract.

A complete classification of the computational complexity of the fixed-point existence problem for Boolean dynamical systems, i.e., finite discrete dynamical systems over the domain {0, 1}, is presented. For function classes \({\mathcal{F}}\) and graph classes \({\mathcal{G}}\), an (\({\mathcal{F}},{\mathcal{G}}\))-system is a Boolean dynamical system such that all local transition functions lie in \({\mathcal{F}}\) and the underlying graph lies in \({\mathcal{G}}\). Let \({\mathcal{F}}\) be a class of Boolean functions which is closed under composition and let \({\mathcal{G}}\) be a class of graphs which is closed under taking minors. The following dichotomy theorems are shown: (1) If \({\mathcal{F}}\) contains the self-dual functions and \({\mathcal{G}}\) contains the planar graphs, then the fixed-point existence problem for (\({\mathcal{F}},{\mathcal{G}}\))-systems with local transition function given by truth-tables is NP-complete; otherwise, it is decidable in polynomial time. (2) If \({\mathcal{F}}\) contains the self-dual functions and \({\mathcal{G}}\) contains the graphs having vertex covers of size one, then the fixed-point existence problem for (\({\mathcal{F}},{\mathcal{G}}\))-systems with local transition function given by formulas or circuits is NP-complete; otherwise, it is decidable in polynomial time.

Mathematics Subject Classification (2000).

68Q17 68Q80 68Q85 68R10 

Keywords.

Discrete dynamical systems fixed points algorithms and complexity 

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Copyright information

© Springer 2008

Authors and Affiliations

  1. 1.Fakultät für InformatikTechnische Universität MünchenGarchingGermany

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